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Porubov A.V.,Institute of Problems in Mechanical Engineering | Andrianov I.V.,RWTH Aachen | Danishevs'Kyy V.V.,PrydniprovSka State Academy of Civil Engineering and Architecture
International Journal of Solids and Structures | Year: 2012

Nonlinear strain wave propagation along the lamina of a periodic two-component composite was studied. A nonlinear model was developed to describe the strain dynamics. The model asymptotically satisfies the boundary conditions between the lamina, in contrast to previously developed models. Our model reduces an initial two-dimensional problem into a single one-dimensional nonlinear governing equation for longitudinal strains in the form of the Boussinesq equation. The width of the lamina may control the propagation of either compression or tensile localized strain waves, independent of the elastic constants of the materials of the composite.© 2012 Elsevier Ltd. All rights reserved.


Andrianov I.V.,RWTH Aachen | Danishevs'Kyy V.V.,Prydniprovska State Academy of Civil Engineering and Architecture | Kalamkarov A.L.,Dalhousie University
International Journal of Solids and Structures | Year: 2012

Static and dynamic problems for the elastic plates and membranes periodically perforated by holes of different shapes are solved using the combination of the singular perturbation technique and the multi-scale asymptotic homogenization method. The problems of bending and vibration of perforated plates are considered. Using the asymptotic homogenization method the original boundary-value problems are reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. In the present paper the perforated plates with large holes are considered, and the singular perturbation method is used to solve the pertinent unit cell problems. Matching of limiting solutions for small and large holes using the two-point Padé approximants is also accomplished, and the analytical expressions for the effective stiffnesses of perforated plates with holes of arbitrary sizes are obtained. © 2011 Elsevier Ltd. All rights reserved.


Andrianov I.V.,RWTH Aachen | Danishevs'Kyy V.V.,Prydniprovska State Academy of Civil Engineering and Architecture | Kalamkarov A.L.,Dalhousie University
International Journal of Solids and Structures | Year: 2012

Static problems for the elastic plates and rods periodically perforated by small holes of different shapes are solved using the asymptotic approach based on the combination of the asymptotic technique and the multi-scale homogenization method. Using the asymptotic homogenization method the original boundary-value problem is reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. The combination of the perturbation method and the technique of successive approximations is applied for the solution of the unit cell problems. Taking into the account small size of holes the method of perturbation of the shape of the boundary and the Schwarz alternating method are used. The problems of torsion of a rod with perforated cross-section; deflection of the perforated membrane loaded by a normal load; and bending of perforated plates with circular and square holes are considered consecutively. The error of the applied asymptotic techniques is estimated and the high accuracy of the obtained solutions is demonstrated. © 2011 Elsevier Ltd. All rights reserved.


Karasev A.G.,PrydniprovsKa State Academy of Civil Engineering and Architecture
Mathematics and Mechanics of Solids | Year: 2014

The influence of periodical imperfections on the buckling load value of shallow elastic closed conical shells undergoing external pressure was investigated by numerical analysis. Results are compared with experimental data. It is established that initial imperfections significantly influence the buckling pressure value only for large imperfection amplitude (an order of magnitude more than the shell thickness). The existence is established of the effect of a 'static resonance'. © The Author(s) 2014.


Andrianov I.V.,RWTH Aachen | Danishevs'kyy V.V.,Prydniprovska State Academy of Civil Engineering and Architecture | Ryzhkov O.I.,Prydniprovska State Academy of Civil Engineering and Architecture | Weichert D.,RWTH Aachen
Wave Motion | Year: 2013

Wave propagation in nonlinear elastic media with microstructure is studied. As an illustrative example, a 1D model of a layered composite material is considered. Geometrical nonlinearity is described by the Cauchy-Green strain tensor. For predicting physical nonlinearity the expression of the energy of deformation as a series expansion in powers of the strains is used. The effective wave equation is derived by the higher-order asymptotic homogenization method. An asymptotic solution of the nonlinear cell problem is obtained using series expansions in powers of the gradients of displacements. Analytical expressions for the effective moduli are presented. The balance between nonlinearity and dispersion results in formation of stationary nonlinear waves that are described explicitly in terms of elliptic functions. In the case of weak nonlinearity, an asymptotic solution is developed. A number of nonlinear phenomena are detected, such as generation of higher-order modes and localization. Numerical results are presented and practical significance of the nonlinear effects is discussed. © 2012 Elsevier B.V.


Andrianov I.V.,RWTH Aachen | Danishevs'Kyy V.V.,Prydniprovska State Academy of Civil Engineering and Architecture | Kalamkarov A.L.,Dalhousie University
Nonlinear Dynamics | Year: 2013

The phenomenon of vibration localization plays an important role in the dynamics of inhomogeneous and nonlinear materials and structures. The vibration localization can occur in the case of inhomogeneity under the following conditions: (i) the frequency spectrum of the periodic structure includes stopbands, (ii) a perturbation of periodicity is present, and (iii) the eigenfrequency of the perturbed element falls into a stopband. Under these conditions, the energy can be spatially localized in the vicinity of the defect with an exponential decay in the infinity. The influence of nonlinearity can shift frequency into the stopband zone. In the present paper, the localization of vibrations in one-dimensional linear and nonlinear lattices is investigated. The localization frequencies are determined and the attenuation factors are calculated. Discrete and continuum models are developed and compared. The limits of the applicability of the continuum models are established. Analysis of the linear problem has allowed a better understanding of specifics of the nonlinear problem and has led to developing a new approach for the analysis of nonlinear lattices alternative to the method of continualization. © 2012 Springer Science+Business Media Dordrecht.


Andrianov I.V.,RWTH Aachen | Danishevs'Kyy V.V.,PrydniprovSka State Academy of Civil Engineering and Architecture | Kalamkarov A.L.,Dalhousie University
Composites Part B: Engineering | Year: 2010

The analytical interpolation formulas for the effective electrical conductivity of fiber-reinforced and particulate composites for any volume fractions of inclusions are derived in the present paper. The Garnett formulas are used for a limiting case of small volume fractions of inclusions. And the formulas based on the Shklovskii-De Gennes model are adopted for a limiting case of large volume fractions of inclusions approaching the percolation threshold. The derivation is based on application of the method of asymptotically equivalent functions. This approach presents a natural generalization of the two-point Padé approximants to the case, when in one of the limits the interpolated function cannot be represented in the form of power series. The application of this interpolation technique made it possible to derive the formulas for the effective conductivity of fiber-reinforced and particulate composites with very high volume fractions of inclusions, up to the percolation threshold. The numerical results are compared with the known asymptotic expressions, and also with the existing exact expressions in some limiting cases, for example, in the case of statistically equivalent arrangement of the constituent materials. Comparison with the experimental data confirms the satisfactory accuracy of the obtained analytical results. © 2010 Elsevier Ltd.


Andrianov I.V.,RWTH Aachen | Danishevs'kyy V.V.,Prydniprovska State Academy of Civil Engineering and Architecture | Ryzhkov O.I.,Prydniprovska State Academy of Civil Engineering and Architecture | Weichert D.,RWTH Aachen
Wave Motion | Year: 2014

Propagation of nonlinear strain waves through a layered composite material is considered. The governing macroscopic wave equation for the long-wave case was obtained earlier by the higher-order asymptotic homogenization method (Andrianov etal., 2013). Non-stationary dynamic processes are investigated by a pseudo-spectral numerical procedure. The time integration is performed by the Runge-Kutta method; the approximation with respect to the spatial co-ordinate is provided by the Fourier series expansion. The convergence of the Fourier series is substantially improved and the Gibbs-Wilbraham phenomenon is reduced with the help of Padé approximants. As result, we explore how fast and under what conditions the solitary strain waves can be generated from an initial excitation. The numerical and analytical solutions (when the latter can be obtained) are in good agreement. © 2013 Elsevier B.V.


Tychynin V.,Prydniprovska State Academy of Civil Engineering and Architecture
Symmetry | Year: 2015

Additional nonlocal symmetries of diffusion-convection equations and the Burgers equation are obtained. It is shown that these equations are connected via a generalized hodograph transformation and appropriate nonlocal symmetries arise from additional Lie symmetries of intermediate equations. Two entirely different techniques are used to search nonlocal symmetry of a given equation: The first is based on usage of the characteristic equations generated by additional operators, another technique assumes the reconstruction of a parametrical Lie group transformation from such operator. Some of them are based on the nonlocal transformations that contain new independent variable determined by an auxiliary differential equation and allow the interpretation as a nonlocal transformation with additional variables. The formulae derived for construction of exact solutions are used. © 2015 by the authors.


Sokolovsky S.A.,Prydniprovska State Academy of Civil Engineering and Architecture
Theoretical and Mathematical Physics | Year: 2011

In the framework of the Bogoliubov method of reduced description of nonequilibrium states, which is based on his functional hypothesis, we obtain a kinetic equation for arbitrary inhomogeneous electron states in a polar crystal in the presence of a strong electric field. Using the reduced description method, we study the equalization of velocities and temperatures of the polaron gas with a small density and the phonon subsystem in the presence of a weak electric field in the spatially homogeneous case. We establish that the nonequilibrium distribution function of a polaron differs from the Maxwell distribution even in the approximation that is linear in small differences between the subsystem velocities and temperatures. We calculate the corresponding relaxation times and polaron mobility. © 2011 Pleiades Publishing, Ltd.

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